Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures

A method is given of evaluating Madelung constants for invariant cubic lattice complexes. It is shown that a linear combination of at most 8 lattice sums is sufficient to give the Madelung constant of all such lattices. In many cases this reduces to just 3 sums. The 8 sums suffice also to determine the Madelung constant for tetragonal lattices with axial ratio square root 2 or 2 and for orthorhombic lattices with lattice parameters in the ratio 1: square root 2:2. The method developed here is also applied to the evaluation of lattice sums arising from particles interacting with a general r -s potential.